Chapter 4 Probability
Example: Multiplication Counting Rule (1 of 2)
When making random guesses for an unknown four- digit passcode, each digit can be 0, 1. . . , 9. What is the total number of different possible passcodes? Given that all of the guesses have the same chance of being correct, what is the probability of guessing the correct passcode on the first attempt?
Example 2: Complementary Events (1 of 2)
A certain group of women has a 0.18% rate of red/green color blindness. If a woman is randomly selected, what is the probability that she does not have red/green color blindness? Solution P(she does not have red/green color blindness) = 100% − 0.18% = 99.82% The probability that she does not have red/green color blindness is 0.9982.
Events, Simple Events, and Sample Spaces.
An event is any collection of results or outcomes of a procedure. • A simple event is an outcome or an event that cannot be further broken down into simpler components. • The sample space for a procedure consists of all possible simple events. That is, the sample space consists of all outcomes that cannot be broken down any further.
Example 1: Complementary Events (1 of 2)
Based on a journal article, the probability of randomly selecting someone who has sleepwalked is 0.292, so P(sleepwalked) = 0.292. If a person is randomly selected, find the probability of getting someone who has not sleepwalked.
Formula separate.
Classical Approach to Probability (Requires Equally Likely Outcomes) If a procedure has n different simple events that are equally likely, and if event A can occur in s different ways, then. Classical Approach Classical Approach to Probability (Requires Equally Likely Outcomes) If a procedure has n different simple events that are equally likely, and if event A can occur in s different ways, then Caution When using the classical approach, always confirm that the outcomes are equally likely.
Relative Frequency Approximation of Probability
Conduct (or observe) a procedure and count the number of times that event A occurs. P(A) is then approximated as follows:
Chapter 4.2
Elementary Statistics Thirteenth Edition Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Chapter 4 Probability Chapter 4: Probability 4.1 Basic Concepts of Probability 4.2 Addition Rule and Multiplication Rule 4.3 Complements and Conditional Probability Concepts Part 1: The Addition Rule • Disjoint versus Not Disjoint • Addition Rule Key Concept In this section we present the addition rule as a tool for finding P(A or B), which is the probability that either event A occurs or event B occurs (or they both occur) as the single outcome of a procedure. The word "or" in the addition rule is associated with the addition of probabilities. Addition Rule for P(A or B): The word or suggests addition, and when adding P(A) and P(B), we must add in such a way that every outcome is counted only once. Compound Event • Compound Event - A compound event is any event combining two or more simple events. Disjoint Events • Disjoint (or mutually exclusive) - Events A and B are disjoint (or mutually exclusive) if they cannot occur at the same time. (That is, disjoint events do not overlap.) Example: Disjoint Events Disjoint events: Event A: Randomly selecting a movie to watch in the theater that is rated PG. Event B: Randomly selecting a movie to watch in the theater that is rated R. The events are disjoint (or mutually exclusive) because the selected movie cannot be both. Example: Not Disjoint Events Events that are not disjoint: Event A: Randomly selecting someone taking a statistics course at the local community college. Event B: Randomly selecting someone that is married. The events are not disjoint because the selected person could be both. Your Turn: Disjoint or Not Disjoint Are the events disjoint? Event A: Randomly choosing a person that plays soccer. Event B: Randomly choosing a person that plays softball. Are the events disjoint? (using a standard 52-card deck) Event A: Randomly selecting a card that is a heart. Event B: Randomly selecting a card that is a spade. Addition Rule Notation P(A or B) = P(in a single trial, event A occurs or event B occurs or they both occur) Formal Addition Rule P(A or B) = P(A) + P(B) − P(A and B) where P(A and B) denotes the probability that A and B both occur at the same time as an outcome in a trial of a procedure. Your Turn 1: Formal Addition Rule Find the probability of randomly selecting a middle school student that plays football or plays baseball, given the probability of randomly selecting a middle school student that plays football is 20%, selecting one that plays baseball is 15%, and selecting one that plays both football and baseball is 10%. Intuitive Addition Rule To find P(A or B), add the number of ways event A can occur and the number of ways event B can occur, but add in such a way that every outcome is counted only once. P(A or B) is equal to that sum, divided by the total number of outcomes in the sample space. Example: Addition Rule Using Tables The table summarized blood group and Rh types for 200 subjects. Find the probability of randomly selecting a subject that is blood group B or type Rh-. Blank O A B AB Rh+ 78 70 16 8 Rh- 12 10 4 2 Solution (using the intuitive approach) (16 + 4 + 12 + 10 + 2) / 200 = 44/200 = 0.22 Your Turn 1: Addition Rule Using Tables The table summarized blood group and Rh types for 200 subjects. Find the probability of randomly selecting a subject that is blood group O or type Rh+. Blank O A B AB Rh+ 78 70 16 8 Rh- 12 10 4 2 Your Turn 2: Addition Rule Using Tables The table summarized blood group and Rh types for 200 subjects. Find the probability of randomly selecting a subject that is blood group O or blood group AB. Blank O A B AB Rh+ 78 70 16 8 Rh- 12 10 4 2 Your Turn 3: Addition Rule Using Tables Use the following results from a test for drug use, which is provided by a certain drug testing company. Among 147 subject with positive test results, there are 23 false positive results; among 159 negative results, there are 4 false negative results. If one of the test subjects is randomly selected, find the probability that the subject tested negative or did not use drugs. Your Turn 1: Addition Rule One card is drawn from a standard 52 card deck. Find the probability that a diamond or red card is drawn. Your Turn 2: Addition Rule One card is drawn from a standard 52 card deck. Find the probability that a jack or black card is drawn. Your Turn 3: Addition Rule A fair 12-sided die is rolled. Find the probability that an odd number or a 4 is rolled. Your Turn 4: Addition Rule A fair 12-sided die is rolled. Find the probability that a number greater than 5 or an even number is rolled. Complementary Events and the Addition Rule Your Turn: Complementary Events and the Addition Rule A fair 6-sided die is rolled. What is the probability that it lands on 4 or does not land on 4? 4.2 Homework - Part 1
Chapter 4.2 part 2
Elementary Statistics Thirteenth Edition Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Chapter 4 Probability Chapter 4: Probability 4.1 Basic Concepts of Probability 4.2 Addition Rule and Multiplication Rule 4.3 Complements and Conditional Probability Concepts Part 2: The Multiplication Rule • Independent versus Dependent • Multiplication Rule • 5% Guideline for Cumbersome Calculations • Redundancy Key Concept This section also presents the basic multiplication rule used for finding P(A and B), which is the probability that event A occurs and event B occurs. The word "and" in the multiplication rule is associated with the multiplication of probabilities. Multiplication Rule for P(A and B): The word and for two trials suggests multiplication, and when multiplying P(A) and P(B), we must be sure that the probability of event B takes into account the previous occurrence of event A. Compound Event • Compound Event - A compound event is any event combining two or more simple events. Independence • Independent - Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. (Several events are independent if the occurrence of any does not affect the probabilities of the occurrence of the others.) - If A and B are not independent, they are said to be dependent. Sampling In the world of statistics, sampling methods are critically important, and the following relationships hold: • Sampling with replacement: Selections are independent events. • Sampling without replacement: Selections are dependent events. Example: Independent or Dependent (1 of 2) Is Event B dependent or independent of Event A? A: A green ball is drawn from a box with five balls and placed next to the box. B: A red ball is drawn next and placed next to the green one. Solution Dependent. The probability of drawing a given ball on the second draw is affected by what occurred on the first draw. Example: Independent or Dependent (2 of 2) Is Event B dependent or independent of Event A? A: A random guess is made at the correct answer of a multiple choice question. B: A random guess is made at the correct answer of a second multiple choice question. Solution Independent. The probability of guessing the first question correctly does not affect the probability of guessing the second question correctly. Your Turn: Independent or Dependent Is Event B dependent or independent of Event A? A: A fair die is rolled and lands on an even number. B: A fair die is rolled and lands on a 2. Is Event B dependent or independent of Event A? A: Randomly selecting a home owner. B: Randomly selecting a lawn mower owner. Multiplication Rule Notation P(A and B) = P(event A occurs in a first trial and event B occurs in a second trial) P(B | A) represents the probability of event B occurring after it is assumed that event A has already occurred. Formal Multiplication Rule P(A and B) = P(A) P(B | A). If A and B are independent, P(A and B) = P(A) P(B). In general, the probability of any sequence of independent events is simply the product of their corresponding probabilities. Intuitive Multiplication Rule To find the probability that event A occurs in one trial and event B occurs in another trial, multiply the probability of event A by the probability of event B, but be sure that the probability of event B is found by assuming that event A has already occurred. Example: Multiplication Rule (1 of 3) 50 test results are selected. 45 were positive test results and 5 were negative test results. a. If 2 of these 50 subjects are randomly selected with replacement, find the probability the first selected person had a positive test result and the second selected person had a negative test result. b. Repeat part (a) by assuming that the two subjects are selected without replacement. Example: Multiplication Rule (2 of 3) Solution a. With Replacement: First selection (with 45 positive results among 50 total results): Second selection (with 5 negative test results among the same 50 total results): We now apply the multiplication rule as follows: P(1st selection is positive and 2nd is negative) Example: Multiplication Rule (3 of 3) Solution b. Without Replacement: Without replacement of the first subject, the second probability must be adjusted to reflect the fact that the first selection was positive and is not available for the second selection. After the first positive result is selected, we have 49 test results remaining, and 5 of them are negative. P(1st selection is positive and 2nd is negative) Your Turn 1: Multiplication Rule The table summarized blood group and Rh types for 200 subjects. Find the probability of randomly selecting three different subjects with A positive blood. Blank O A B AB Rh+ 78 70 16 8 Rh- 12 10 4 2 Your Turn 2: Multiplication Rule The table summarized blood group and Rh types for 200 subjects. Sampling with replacement, find the probability of randomly selecting three subjects with A positive blood. Blank O A B AB Rh+ 78 70 16 8 Rh- 12 10 4 2 Your Turn 3: Multiplication Rule A quick quiz consists of a true or false question followed by a multiple choice question with four possible answer (a, b, c, or d). An unprepared student makes random guesses for both answers. If each question has only one correct answer, what is the probability the student guesses both answers correctly? Your Turn 4: Multiplication Rule Two cards are drawn randomly (without replacement) from a standard 52-card deck. What is the probability of selecting a Jack followed by an Ace? Your Turn 5: Multiplication Rule A shopper finds a bin of t-shirts containing 4 size large, 7 size medium and 11 size small. If the shopper is looking for a size large, and doesn't find it on the first draw so sets it aside, what is the probability the shopper continues this process and finds the size large on the third random draw from the bin? Your Turn 6: Multiplication Rule In a case in Riverhead, New York, nine different crime victims listened to voice recordings of five different men. All nine victims identified the same voice as that of the criminal. If the voice identifications were made by random guesses, find the probability that all nine victims would select the same person? Does this constitute reasonable doubt? Treating Dependent Events and Independent 5% Guideline for Cumbersome Calculations When sampling without replacement and the sample size is no more than 5% of the size of the population, treat the selections as being independent (even though they are actually dependent). Example: 5% Guideline for Cumbersome Calculations (1 of 3) Assume that three adults are randomly selected without replacement from the 247,436,830 adults in the United States. Also assume that 23% of adults in the United States have a silver car. Find the probability that the three selected adults all have a silver car. Example: 5% Guideline for Cumbersome Calculations (2 of 3) Solution Because the three adults are randomly selected without replacement, the three events are dependent, but here we can treat them as being independent by applying the 5% guideline for cumbersome calculations. The sample size of 3 is clearly no more than 5% of the population size of 247,436,830. Example: 5% Guideline for Cumbersome Calculations (3 of 3) Solution We get P(all 3 adults have a silver car) = P(first silver car and second silver car and third silver car) = P(first silver car) · P(second silver car) · P(third silver car) = (0.23)(0.23)(0.23) = 0.0122 There is a 0.0122 probability that all three selected adults have a silver car. Your Turn: 5% Guideline for Cumbersome Calculations A quality control analyst randomly selects 15 different car ignition systems from a manufacturing process that has just produced 400 systems, including 10 that are defective. Does this process involve independent events? Can the 5% guideline for cumbersome calculations be used here? What is the probability that all 15 ignition systems are good treating the events independently. Redundancy: Important Application of the Multiplication Rule The principle of redundancy is used to increase the reliability of many systems. Our eyes have passive redundancy in the sense that if one of them fails, we continue to see. An important finding of modern biology is that genes in an organism can often work in place of each other. Engineers often design redundant components so that the whole system will not fail because of the failure of a single component. Example: Redundancy (1 of 3) Modern aircraft are now highly reliable, and one design feature contributing to that reliability is the use of redundancy, whereby critical components are duplicated so that if one fails, the other will work. For example, the Airbus 310 twin-engine airliner has three independent hydraulic systems, so if any one system fails, full flight control is maintained with another functioning system. Example: Redundancy (2 of 3) For this example, we will assume that for a typical flight, the probability of a hydraulic system failure is 0.002. Given that the Airbus 310 actually has three independent hydraulic systems, what is the probability that on a typical flight, control won't be maintained with a working hydraulic system? Solution With three independent hydraulic systems, flight control will be not be maintained if all three systems fail. The probability of all three hydraulic systems failing is 0.002 · 0.002 · 0.002 = 0.000000008. Example: Redundancy (3 of 3) Interpretation Your Turn: Redundancy Commercial aircraft used for flying in hazardous conditions must have two independent radios instead of one. Assume that for a typical flight, the probability of a single radio failure is 0.4%. What is the probability that a particular flight will be threatened with the failure of both radios? 4.2 Homework - Part 2
Chapter 4
Elementary Statistics Thirteenth Edition Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Chapter 4 Probability Chapter 4: Probability 4.1 Basic Concepts of Probability 4.2 Addition Rule and Multiplication Rule 4.3 Complements and Conditional Probability Concepts • Basics of Probability • Relative Frequency Approximation of Probability • Classical Approach to Probability • Law of Large Numbers • Complementary Events • Odds Key Concept The single most important objective of this section is to learn how to interpret probability values, which are expressed as values between 0 and 1. A small probability, such as 0.001, corresponds to an event that rarely occurs. Next are odds and how they relate to probabilities. Odds are commonly used in situations such as lotteries and gambling. Notation for Probabilities P denotes a probability. A, B, and C denote specific events. P(A) denotes the "probability of event A occurring." Probability Limits For any event A, the probability of A is between 0 and 1 inclusive: 0 ≤ P(A) ≤ 1. • The probability of an impossible event is 0. • The probability of a event that is certain to occur is 1. Example: Using a standard 6-sided die, find: • P(rolling a number greater than 7) = 0 • P(rolling a number less than 7) = 1 Common Expressions of Likelihood Possible values of probabilities and the more familiar and common expressions of likelihood Unlikely Events Unless stated otherwise, consider an event to be unlikely, unusual, or significant if its probability is less than or equal to 0.05. Rounding Probabilities When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to three significant digits. Approaches to Probability • Relative Frequency Approximation of Probability • Classical Approach to Probability • Subjective Probability - The probability of an event is estimated by using knowledge of the relevant circumstances. • Simulations - Sometimes none of the preceding three approaches can be used. A simulation of a procedure is a process that behaves in the same ways as the procedure itself so that similar results are produced. Probabilities can sometimes be found by using a simulation. Relative Frequency Approach Relative Frequency Approximation of Probability Conduct (or observe) a procedure and count the number of times that event A occurs. P(A) is then approximated as follows: Example 1: Relative Frequency Approach In a recent year, there were about 3,000,000 skydiving jumps and 21 of them resulted in deaths. Find the probability of dying when making a skydiving jump. Solution We use the relative frequency approach as follows: Example 2: Relative Frequency Approach (1 of 2) In a study of U.S. high school drivers, it was found that 3785 texted while driving during the previous 30 days, and 4720 did not text while driving during that same time period. Based on these results, if a high school driver is randomly selected, find the probability that he or she texted while driving during the previous 30 days. Example 2: Relative Frequency Approach (2 of 2) Solution We can now use the relative frequency approach as follows: There is a 0.445 probability that if a high school driver is randomly selected, he or she texted while driving during the previous 30 days. Your Turn 1: Relative Frequency Approach Use the following table to find the probability of a false positive. blank Positive Test Result (Test shows has strep throat.) Negative Test Result (Test shows does not have strep throat.) Subject Has Strep Throat 45 (True Positive) 5 (False Negative) Subject Does Not Have Strep Throat 25 (False Positive) 480 (True Negative) Your Turn 2: Relative Frequency Approach Use the following table to find the probability of a negative test result. blank Positive Test Result (Test shows has strep throat.) Negative Test Result (Test shows does not have strep throat.) Subject Has Strep Throat 45 (True Positive) 5 (False Negative) Subject Does Not Have Strep Throat 25 (False Positive) 480 (True Negative) Classical Approach Classical Approach to Probability (Requires Equally Likely Outcomes) If a procedure has n different simple events that are equally likely, and if event A can occur in s different ways, then Caution When using the classical approach, always confirm that the outcomes are equally likely. Events, Simple Events, and Sample Spaces • An event is any collection of results or outcomes of a procedure. • A simple event is an outcome or an event that cannot be further broken down into simpler components. • The sample space for a procedure consists of all possible simple events. That is, the sample space consists of all outcomes that cannot be broken down any further. Example: Events, Simple Events, and Sample Spaces In the following display, we use "h" to denote a head and "t" to denote a tail. Procedure Example of Event Sample Space: Complete List of Simple Events Single toss of a coin 1 tail (t is a simple event) {h, t} 3 tosses of a coin 2 heads and 1 tail (hht, hth, and thh are all simple events resulting in 2 heads and 1 tail) {hhh, hht, hth, htt, thh, tht, tth, ttt} Your Turn: Classical Approach For three tosses of a coin, find the probability of the following: • Getting exactly one tail. •Getting at least 2 tails. •Getting at most 2 tails. •Getting all tails. Multiplication Counting Rule Fundamental Counting Principle - For a sequence of events in which the first event can occur n1 ways, the second event can occur n2 ways, the third event can occur n3 ways, and so on, the total number of outcomes is n1 · n2 · n3 . . . . Example: Multiplication Counting Rule (1 of 2) When making random guesses for an unknown four- digit passcode, each digit can be 0, 1. . . , 9. What is the total number of different possible passcodes? Given that all of the guesses have the same chance of being correct, what is the probability of guessing the correct passcode on the first attempt? Example: Multiplication Counting Rule (2 of 2) Solution There are 10 different possibilities for each digit, so the total number of different possible passcodes is n1 · n2 · n3 · n4 = 10 · 10 · 10 · 10 = 10,000. The size of the sample space will be 10,000. Of the 10,000 equally likely passcodes, there is 1 passcode that is correct. The probability of guessing the correct passcode on the first attempt is 1 out of 10,000 or 0.0001. The Law of Large Numbers • Law of Large Numbers - As a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability. CAUTIONS 1. The law of large numbers applies to behavior over a large number of trials, and it does not apply to any one individual outcome. 2. If we know nothing about the likelihood of different possible outcomes, we should not assume that they are equally likely. The actual probability depends on factors such as the amount of preparation and the difficulty of the test. Complement • Complement • The complement of an event A may also be denoted by A' with a notation of P(A') indicating the probability that event A does not occur. Rule of Complementary Events Example 1: Complementary Events (1 of 2) Based on a journal article, the probability of randomly selecting someone who has sleepwalked is 0.292, so P(sleepwalked) = 0.292. If a person is randomly selected, find the probability of getting someone who has not sleepwalked. Example 1: Complementary Events (2 of 2) Solution Using the rule of complementary events, we get P(has not sleepwalked) = 1 − P(sleepwalked) = 1 − 0.292 = 0.708 The probability of randomly selecting someone who has not sleepwalked is 0.708. Example 2: Complementary Events (1 of 2) A certain group of women has a 0.18% rate of red/green color blindness. If a woman is randomly selected, what is the probability that she does not have red/green color blindness? Example 2: Complementary Events (2 of 2) Solution P(she does not have red/green color blindness) = 100% − 0.18% = 99.82% The probability that she does not have red/green color blindness is 0.9982. Actual Odds • Actual Odds Against • Actual Odds in Favor Your Turn: Odds to Probability • If the odds against an event occurring are 6:1, what is the probability the event will occur? • If the odds in favor of an event occurring are 4:5, what is the probability the event will occur? Your Turn: Probability to Odds • If the probability an event will not occur is 3/8, what are the odds against the event occurring? • If the probability an event will occur is 40%, what are the odds in favor of that event occurring? Payoff Odds • Payoff Odds - The payoff odds against event A occurring are the ratio of net profit (if you win) to the amount bet: Payoff odds against event A = (net profit):(amount bet) Your Turn: Payoff Odds Calculate the payoff odds against an event when a $800 bet resulted in a return of $16,800 upon winning. Example: Actual Odds Versus Payoff Odds (1 of 4) a. Find the actual odds against the outcome of 13. b. How much net profit would you make if you win by betting $5 on 13? c. If the casino was not operating for profit and the payoff odds were changed to match the actual odds against 13, how much would you win if the outcome were 13? Example: Actual Odds Versus Payoff Odds (2 of 4) Solution Example: Actual Odds Versus Payoff Odds (3 of 4) Solution b. Because the casino payoff odds against 13 are 35:1, we have 35:1 = (net profit):(amount bet) So there is a $35 profit for each $1 bet. For a $5 bet, the net profit is $175 (which is 5 * $35). The winning bettor would collect $175 plus the original $5 bet. After winning, the total amount collected would be $180, for a net profit of $175. Example: Actual Odds Versus Payoff Odds (4 of 4) Solution c. If the casino were not operating for profit, the payoff odds would be changed to 37:1, which are the actual odds against the outcome of 13. With payoff odds of 37:1, there is a net profit of $37 for each $1 bet. For a $5 bet, the net profit would be $185. (The casino makes its profit by providing a profit of only $175 instead of the $185 that would be paid with a roulette game that is fair instead of favoring the casino.) 4.1 Homework
Chapter 4.3
Elementary Statistics Thirteenth Edition Copyright © 2018, 2014, 2012 Pearson Education, Inc. All Rights Reserved Chapter 4 Probability Chapter 4: Probability 4.1 Basic Concepts of Probability 4.2 Addition Rule and Multiplication Rule 4.3 Complements and Conditional Probability Concepts • Conditional Probabilities • Complements Involving "At Least One" Key Concept We extend the use of the multiplication rule to include the probability that among several trials, we get at least one of some specified event. We consider conditional probability: the probability of an event occurring when we have additional information that some other event has already occurred. Conditional Probability • Conditional Probability - A conditional probability of an event is a probability obtained with the additional information that some other event has already occurred. Notation P(B | A) denotes the conditional probability of event B occurring, given that event A has already occurred. Formal Approach for Finding P(B | A) The probability P(B | A) can be found by dividing the probability of events A and B both occurring by the probability of event A: Intuitive Approach for Finding P(B | A) The conditional probability of B occurring given that A has occurred can be found by assuming that event A has occurred and then calculating the probability that event B will occur. Example: Conditional Probability (1 of 5) Find the following using the table: a. If 1 of the 555 test subjects is randomly selected, find the probability that the subject had a positive rapid strep test, given that the subject actually had strep throat. That is, find P(positive test result | subject had the illness). blank Positive Test Result (Test shows has strep throat.) Negative Test Result (Test shows does not have strep throat.) Subject Has Strep Throat 45 (True Positive) 5 (False Negative) Subject Does Not Have Strep Throat 25 (False Positive) 480 (True Negative) Example: Conditional Probability (2 of 5) Solution a. Intuitive Approach: We want P(positive test result | subject has strep throat). If we assume that the selected subject actually has strep throat, we are dealing only with the 50 subjects in the first row of the table. Among those 50 subjects, 45 had positive test results, so we get this result: P(positive test result | subject has strep throat) Example: Conditional Probability (3 of 5) Find the following using the table: b. If 1 of the 555 test subjects is randomly selected, find the probability that the subject actually has strep throat, given that he or she had a positive test result. That is, find P(subject has strep throat | positive test result). blank Positive Test Result (Test shows has strep throat.) Negative Test Result (Test shows does not have strep throat.) Subject Has Strep Throat 45 (True Positive) 5 (False Negative) Subject Does Not Have Strep Throat 25 (False Positive) 480 (True Negative) Example: Conditional Probability (4 of 5) Solution b. Intuitive Approach: Here we want P(subject has strep throat | positive test result). This is the probability that the selected subject has strep throat, given that the subject had a positive test result. If we assume that the subject had a positive test result, we are dealing with the 70 subjects in the first column of the table. Among those 70 subjects, 45 have strep throat, so P(subject has strep throat | positive test result) Example: Conditional Probability (5 of 5) Interpretation The first result of P(positive test result | subject has strep throat) = 0.900 indicates that a subject who has strep throat has a 0.900 probability of getting a positive test result. The second result of P(subject has strep throat | positive test result) = 0.643 indicates that for a subject who gets a positive test result, there is a 0.643 probability that this subject actually has strep throat. Note that P(positive test result | subject has strep throat) ≠ P(subject has strep throat | positive test result). Confusion of the Inverse In the prior example, P(positive test result | subject has strep throat) ≠ P(subject has strep throat | positive test result). This example proves that in general, P(B | A) ≠ P(A | B). (There could be individual cases where P(A | B) and P(B | A) are equal, but they are generally not equal.) To incorrectly think that P(B | A) and P(A | B) are equal or to incorrectly use one value in place of the other is called confusion of the inverse. Your Turn 1: Conditional Probability Find the probability that the toss of a fair 12-sided die lands on a 2, given that the number it lands on is even. Find the probability that a randomly selected face card is a Queen, given that the card is a face (or court) card. Your Turn 2: Conditional Probability The table summarized blood group and Rh types for 200 subjects. Find the probability of randomly selecting a subject with group A blood given that they are type Rh-. Blank O A B AB Rh+ 78 70 16 8 Rh- 12 10 4 2 Your Turn 3: Conditional Probability The table summarized blood group and Rh types for 200 subjects. Find the probability of randomly selecting a subject with type Rh- given that the subject is known to have group A blood. Blank O A B AB Rh+ 78 70 16 8 Rh- 12 10 4 2 Your Turn 4: Conditional Probability The table summarized blood group and Rh types for 200 subjects. Find the probability of randomly selecting a subject with group AB blood given that they are type Rh+. Blank O A B AB Rh+ 78 70 16 8 Rh- 12 10 4 2 Complements: The Probability of "At Least One" When finding the probability of some event occurring "at least once," we should understand the following: • "At least one" has the same meaning as "one or more." • The complement of getting "at least one" particular event is that you get no occurrences of that event. The Probability of "At Least One" Finding the probability of getting at least one of some event: 1. Let A = getting at least one of some event. 4. Subtract the result from 1. That is, evaluate this expression: P(at least one occurrence of event A) = 1 − P(no occurrences of event A) Example: The Probability of "At Least One" (1 of 3) A study by SquareTrade found that 6% of damaged iPads were damaged by "bags/backpacks." If 20 damaged iPads are randomly selected, find the probability of getting at least one that was damaged in a bag/backpack. Example: The Probability of "At Least One" (2 of 3) Solution Step 1: Let A = at least 1 of the 20 damaged iPads was damaged in a bag/backpack. Example: The Probability of "At Least One" (3 of 3) Solution Your Turn 1: The Probability of "At Least One" Commercial aircraft used for flying in hazardous conditions must have two independent radios instead of one. Assume that for a typical flight, the probability of a single radio failure is 0.4%. What is the probability that a particular flight will have at least one working radio? Your Turn 2: The Probability of "At Least One" The Orange County Department of Public Health test water for contamination due to the presence of E. coli bacteria. To reduce laboratory costs, water samples from six public swimming areas are combined for one test, and further testing is done only if the combined sample fails. Based on past results, there is a 2% chance of finding E. coli bacteria in a public swimming area. Find the probability that a combined sample from six public swimming areas will reveal the presence of E. coli bacteria. Your Turn 3: The Probability of "At Least One" The FICO (Fair Isaac & Company) score is commonly used as a credit rating. There is a 1% delinquency rate among consumers who have a FICO score above 800. If four consumers with FICO scores above 800 are randomly selected, find the probability that at least one of them becomes delinquent. 4.3 Homework Chapter 4 Review Homework Quiz 4 Chapter 4 EXAM
Probability Limits
For any event A, the probability of A is between 0 and 1 inclusive: 0 ≤ P(A) ≤ 1. • The probability of an impossible event is 0. • The probability of a event that is certain to occur is 1. Example: Using a standard 6-sided die, find: • P(rolling a number greater than 7) = 0 • P(rolling a number less than 7) = 1
Your Turn: Classical Approach.
For three tosses of a coin, find the probability of the following: • Getting exactly one tail. •Getting at least 2 tails. •Getting at most 2 tails. •Getting all tails.
Multiplication Counting Rule.
Fundamental Counting Principle - For a sequence of events in which the first event can occur n1 ways, the second event can occur n2 ways, the third event can occur n3 ways, and so on, the total number of outcomes is n1 · n2 · n3 .
Example #1
In a recent year, there were about 3,000,000 skydiving jumps and 21 of them resulted in deaths. Find the probability of dying when making a skydiving jump. Solution We use the relative frequency approach as follows:
Example # 2
In a study of U.S. high school drivers, it was found that 3785 texted while driving during the previous 30 days, and 4720 did not text while driving during that same time period. Based on these results, if a high school driver is randomly selected, find the probability that he or she texted while driving during the previous 30 days.
Example: Events, Simple Events, and Sample Spaces.
In the following display, we use "h" to denote a head and "t" to denote a tail. Procedure Example of Event Sample Space: Complete List of Simple Events Single toss of a coin 1 tail (t is a simple event) {h, t} 3 tosses of a coin 2 heads and 1 tail (hht, hth, and thh are all simple events resulting in 2 heads and 1 tail) {hhh, hht, hth, htt, thh, tht, tth, ttt}.
The Law of Large Numbers.
Law of Large Numbers - As a procedure is repeated again and again, the relative frequency probability of an event tends to approach the actual probability. CAUTIONS 1. The law of large numbers applies to behavior over a large number of trials, and it does not apply to any one individual outcome. 2. If we know nothing about the likelihood of different possible outcomes, we should not assume that they are equally likely. The actual probability depends on factors such as the amount of preparation and the difficulty of the test.
Basics of Probability
Notation for Probabilities P denotes a probability. A, B, and C denote specific events. P(A) denotes the "probability of event A occurring."
rule of complementary events
P(A) + P(Ā) = 1 P(Ā) = 1 - P(A) P(A) = 1 - P(Ā)
Common Expressions of Likelihood
Possible values of probabilities and the more familiar and common expressions of likelihood.
Example 1: Relative Frequency Approach.
Rounding Probabilities When expressing the value of a probability, either give the exact fraction or decimal or round off final decimal results to three significant digits.
Example: Multiplication Counting Rule.
Solution There are 10 different possibilities for each digit, so the total number of different possible passcodes is n1 · n2 · n3 · n4 = 10 · 10 · 10 · 10 = 10,000. The size of the sample space will be 10,000. Of the 10,000 equally likely passcodes, there is 1 passcode that is correct. The probability of guessing the correct passcode on the first attempt is 1 out of 10,000 or 0.0001.
Example 1: Complementary Events (2 of 2)
Solution Using the rule of complementary events, we get P(has not sleepwalked) = 1 − P(sleepwalked) = 1 − 0.292 = 0.708 The probability of randomly selecting someone who has not sleepwalked is 0.708.
Example #2
Solution We can now use the relative frequency approach as follows: There is a 0.445 probability that if a high school driver is randomly selected, he or she texted while driving during the previous 30 days.
Key Concepts
The single most important objective of this section is to learn how to interpret probability values, which are expressed as values between 0 and 1. A small probability, such as 0.001, corresponds to an event that rarely occurs. Next are odds and how they relate to probabilities. Odds are commonly used in situations such as lotteries and gambling.
Unlikely Events
Unless stated otherwise, consider an event to be unlikely, unusual, or significant if its probability is less than or equal to 0.05.
Your Turn 1: Relative Frequency Approach.
Use the following table to find the probability of a false positive.
Your Turn 2: Relative Frequency Approach.
Use the following table to find the probability of a negative test result.
Approaches to Probability.
• Relative Frequency Approximation of Probability • Classical Approach to Probability • Subjective Probability - The probability of an event is estimated by using knowledge of the relevant circumstances. • Simulations - Sometimes none of the preceding three approaches can be used. A simulation of a procedure is a process that behaves in the same ways as the procedure itself so that similar results are produced. Probabilities can sometimes be found by using a simulation.
Compliment
• The complement of an event A may also be denoted by A' with a notation of P(A') indicating the probability that event A does not occur.
